I apologize if this is supposed to be obvious, but ... . Libor spot rates are quoted up to a year, beyond that one can use Eurodollar futures to continue to build the curve. Let's say up to 3 years. Beyond that one can ... well that is what I don't know. Google says "benchmark swaps", but that is the "chicken or egg" problem. Presumbably swap rates can be derived from Libor forward rates and vice versa. But where are they coming from? It can't be Treasuries due to lack of AA credit risk. It can't be corporates. They're not quoted with the BBA (unless I'm wrong). So if you had to trade the first and only swap in the world, where would you get the swap rate for say 5 to 30 years?
The short answer is that Libor swap rates come from the market. They represent a series of cashflows in the future whose value is determined by the fixing, which the market participants have their own valuations of.
Since the actual cash flows are now discounted using a separate funding curve, the swap prices embed both a prediction of future fixings and a measure of counterparty risk.
Libor no longer represents the actual cost of funding in the market, so Libor instruments involve buying or selling exposure to this Libor fixing index, which is now only correlated to funding costs.
You are asking for the arbitrage path/replication basket, for which there is only a series of FRAs funded with your funding curve - nothing else will give you the Libor exposure you need in the resulting basket.
Swaps are the market's medium to long term interest rate instrument.